WEBVTT 00:00:02.720 --> 00:00:09.530 Given the 𝐴𝐡𝐢𝐷𝐸 is a regular pentagon, find the measure of angle 𝐴𝐡𝑋. 00:00:11.590 --> 00:00:15.840 Firstly, let’s mark the angle that we’re looking to find on the diagram. 00:00:17.810 --> 00:00:23.000 𝐴𝐡𝑋 is the angle formed when you travel from 𝐴 to 𝐡 to 𝑋. 00:00:23.180 --> 00:00:25.890 So it’s this angle here, marked in orange. 00:00:27.660 --> 00:00:37.090 We can see that the diagram consists of a triangle, triangle π΅π‘‹π‘Œ, and then the pentagon 𝐴𝐡𝐢𝐷𝐸 which we’re told is regular. 00:00:37.800 --> 00:00:40.600 Let’s think about how we’re going to approach this problem. 00:00:42.310 --> 00:00:55.140 The angle that we’re looking for, angle 𝐴𝐡𝑋, sits on a straight line with two other angles: angle π‘‹π΅π‘Œ inside the triangle and angle 𝐴𝐡𝐢 inside the pentagon. 00:00:56.600 --> 00:01:06.200 If we can work out these two other angles, then we can calculate angle 𝐴𝐡𝑋 using the fact that angles on a straight line sum to 180 degrees. 00:01:08.020 --> 00:01:10.640 Let’s think about the angle in the triangle first of all. 00:01:11.290 --> 00:01:15.340 Remember, the angle sum in a triangle is always 180 degrees. 00:01:15.570 --> 00:01:20.270 And as we’ve been given the measures of the other two angles, we can calculate the third. 00:01:22.030 --> 00:01:30.630 So angle π‘‹π΅π‘Œ is 180 degrees minus 79 degrees minus 64 degrees which is 37 degrees. 00:01:32.360 --> 00:01:36.610 Next, let’s think about the angle in the pentagon, angle 𝐴𝐡𝐢. 00:01:38.180 --> 00:01:50.850 A key fact about polygons is that the sum of their interior angles can be calculated by multiplying 180 by 𝑛 minus two, where 𝑛 represents the number of sides in the polygon. 00:01:52.470 --> 00:01:55.360 Our polygon is a pentagon which has five sides. 00:01:55.570 --> 00:02:02.680 Therefore, the sum of its interior angles is found by multiplying 180 by three which is 540. 00:02:04.380 --> 00:02:10.270 Now this is the sum of all of the interior angles in the pentagon, not the size of each individual angle. 00:02:11.800 --> 00:02:17.830 The key piece of information given in the question is that 𝐴𝐡𝐢𝐷𝐸 is a regular pentagon. 00:02:18.000 --> 00:02:21.700 Which means that all of its interior angles are the same size. 00:02:23.340 --> 00:02:28.190 Therefore, each interior angle can be found by dividing the sum by five. 00:02:28.600 --> 00:02:33.140 540 divided by five which is 108 degrees. 00:02:35.110 --> 00:02:40.800 So now we know the size of angle 𝐴𝐡𝐢 and the size of angle π‘‹π΅π‘Œ. 00:02:41.400 --> 00:02:46.960 Remember, these two angles sit on a straight line with angle 𝐴𝐡𝑋, which we’re looking to calculate. 00:02:48.740 --> 00:02:56.540 So angle 𝐴𝐡𝑋 is equal to 180 degrees minus 108 degrees minus 37 degrees. 00:02:58.290 --> 00:03:02.540 The measure of angle 𝐴𝐡𝑋 is 35 degrees.